The  Lunar  Apsides 


COLBERT. 


c 


Is  i    KniTiox,    100  COPIES. 


MOTION    OF    THE    LUNAR    A.VSLDKS. 


The  motion  of  the  lunar  orbit  lias  long  been  a  vexed  question 
with  the  mathematicians.  The  following  method  of  reconciling 
the  theory  with  observation  is  novel,  and  perhaps  will  be  accepted 
as  conclusive : 

It  may  be  mentioned,  incidentally,  that  the  motion  of  the  lunar 
apsides  has  for  the  last  two  hundred  years  been  a  stumbling- 
block.  NEWTON  tried  to  account  for  it  011  the  gravitation  theory, 
but  left  it  with  the  remark,  "Apsis  lumr  est  duplo  velocior 
circiter"  (the  motion  is  about  twice  as  great  as  this).  CLAIRAUT 
showed,  in  1750,  how  to  account  theoretically  for  the  other  half, 
but  the  attempt  to  reduce  the  equations  to  a  numerical  form  still 
left  a  residual,  and  when  LAPLACE  attacked  the  problem  he  was 
only  able  to  make  the  theory  responsible  for  444  parts  out  of 
445.  It  has  been  attempted  to  bridge  over  the  difficulty  by  adding 
2  sin'2  | -y  .  d  Q>  -f-  d  t  to  the  motion  of  the  perihelion ;  which  in 
the  case  of  the  moon  is  practically  equal  to  4  sin4  ^y,  because 
d Q  -4-  d  D  =  -|-  sin2  y;  =2  sin2  \ y,  nearly.  It  will  be 
observed  that  this  quantity  is  not  needed  in  that  shape,  neither 
is  the  existence  of  a  second  moon  required  to  account  for  the  peri- 
geal  motion.  I  may  not  be  familiar  with  all  the  literature  of  the 
subject,  but  believe  that  the  outstanding  residual  has  not  hitherto 
been  eliminated  by  any  investigator;  and  note^  a  recent  remark 
by  G.  W.  HILL  to  the  effect  that  it  is  not  probable  the  perigeal 
motion  will  ever  be  accounted  for  by  theory  so  closely  as  it  can 
be  obtained  by  a  comparison  of  observations. 

If  1  —  c  represent  the  motion  of*  the  perigee  divided  by  that 
of  the  moon,  then  c2  and  (1  —  c"2)  are  the  squares  of  two  sides 
of  the  right-angled  triangle  the  hypothenuse  of  which  is  unity; 
and  v  1  —  c2  is  the  perturbation  of  the  radius  vector.  (This  is 
not  new.) 

The  quantity  1  — c-  comprises  a  radial,  which  involves  r3:  a 
tangential,  depending  on  the  square  of  the  velocity  in  the  orbit, 
involving  r4;  and  one  that  originates  in  the  displacement,  being 
really  a  perturbation  of  the  perturbation.  The  last  is  usually 
treated  as  a  single  quantity,  namely  as  a  function  of  r4.  It  is 
more  philosophical  to  regard  it  as  furnishing  a  multiple  for  each 


736356 


MOTION    OF    THE 


.of'  •*  the  :  |)tliiiiv'ftwo  '  instead  of  being  a  quantity  simply  additive. 
Also,  for  obtaining  the  mean  motion  of  the  apsides,  it  is  sufficient 
to  derive  the  constant  portion  of  each  function  considered,  being 
what  we  shall  here  call  the  "Average  Value"  of  the  quantity: 

With  g  the  mean  anomaly,  e  the  eccentricity,  r  the  radius 
vector,  and  a  the  semi  -axis  major,  we  have  the  following  extension 
of  a  well-known  equation: 


cos  g 


315 

And    an  inversion  of   this  series    gives  the    following,   which  it 
is  not  necessary  to  carry  out  beyond  the  sixth  power : 

-       =      !  •  +     (  +     «  -  —         +       j~-  )  cos  ,, 

y   ..     -»-••  + 


4- 

120 


LUNAH    APSIDES.  3 

liaising  eacli  of  these  expressions  to  the  required  powers,  and 
omitting  all  that  is  periodical,  we  have  the  following  as  "average 
values": 

*>ri  :  i  +  f 
-  ^  i+J'* 

"      '-     :         1    4-    3e-        -f-      3  <H       +      I,,  e«       —      -Ii_   „» 
«s  8  8~  3-83 

,},4  1  ?i  1  J  Q 

«      '__     :         1    +    oe*       -f     ,K)e4     4-    _i±_   c«    _       i__  e8 
«4  8  9-8*  4-8* 


f     «*:        1    4-     *2        +      ?*<        +      15e« 
r2  2  8  48 

«      fl3   :         14.    ?«2       4-      15.^4       +      7     « 
r3  284 

«4  o    o  45    ,  35    „ 

^  :  '8  e  T 

For  the  solution  of  the  problem  we  take  the  following  as 
the  most  probable  values  of  the  quantities  named.  They  are 
deduced  from  the  figures  given  by  NEWCOMB,  in  1879,  in  his 
paper  011  the  recurrence  of  solar  eclipses.  The  epoch  chosen  is 
A.  D.  1800.  The  processes  of  the  subsequent  computation  are 
given,  as  they  may  be  of  use  in  verification;  and  all  of  the  loga- 
rithms have  been  computed  closely  enough  to  secure  accuracy  in 
the  last  figure  of  the  result  as  here  presented.  The  mark  ? 
following  some  of  the  numbers  or  logarithms  indicates  that  the 
next  succeeding  figure  would  be  nearly  5.  The  same  mark 
inverted,  thus,  4  shows  that  the  given  value  is  too  great  by  half 
a  unit,  or  nearly  so,  in  the  right  hand  place: 
'  Log&ri&ms. 

©'s  daily  motion,  3548".1927904  3-5500072091 

Sidereal  year,  days,  365   -2563647  2-56259  77924 

D's  synodical  rev.  days,  29   -53058844  1-4702721009' 

Sidereal  rev.  days,  27   -32166120  1-4365071016 

]>  's  daily  motion,  47434X/.890233  4-67609  78999' 

D  's  —  Q's  daily  motion,       43886".697443  4-64233  29006 

D  's  TT,  daily  motion,  400".9187565  2-60305  63747 


MOTION    OF    THK 


Half  square  ratio  sidereal  periods; 

=  (l  -i-  357-447).   1)7-44678  802:27 
358-447  4-  357-447        '0-00121  32943 


_  =  0-0027898145 

O 


Nominal  perturbation ; 

Then    for    en    the     eccentricity    of     the 

by    a    preceding    formula: 


97-4455753284 
earth's     orbit,    with 


e,     =     0-01679228,    we    have     by    a 

(  a,  4-  r,)3  1-00042  31202  0-00018  37199 

and  taking  an  approximate  value  for   J>  's     c,     with     y     =     about 
5°8/40".6  we  obtain  E,  the    perturbation    of  the    perigeal    motion 
due    to    the    earth's    elliptical    figure,  as    follows : 
Constant  of  precession  (JULIAN)   =   54". 9  4625 
Obliquity,  (1800)   =   28°  27'  54".8  cos 

50".40230 
Daily  soli-lunar 


1-73993  805 
9-96251  23' 
1-70245  04 


])  's    m  -±-  a3    X     © 
D  's  ( a  -4-  r)3 
1  _  (3  4.  2)  sin'2  y 


©'s   (a,-?-  r,)3 

Sum  of  D  and  ©    = 


0".l 379940 
1-00453806 

2-154902 
1-000423 


9-1398602 

0-33673  10 
0-00196  64 
9-99473  024 
0-33342  75 


0.49904  41 
6-28988  37 
94-87480  44 
99-96251  23' 
0-8601398 
2-48638  44 

99-12395  63 
98-97424  36 
98-0981999 
93-42210  20 

0-00001  36383 

Our  value  of  E  is  slightly  larger  than  the  one  given  by  LAPLACE. 
The  precession  here  used  is  greater  than  that  observed ;  the  differ- 
ence being  due  to  a  planetary  perturbation  which  causes  the 
equinox  to  move  forward  a  little  more  than  17"  in  a  century. 
The  number  306.468  is  the  earth's  moment  of  inertia,  divided 
by  the  momentum  of  the  ring  of  matter  that  forms  our  equator- 
ial protuberance. 


3-155325 

(3  -J-  2)  seconds  sidereal  arc  in  solar  day, 
(Solar  days  in  sidereal  year)-  a.  c. 

Obliquity  of  ecliptic,  cos 

Daily  soli -lunar  precession,     0".  13799  40   a.  c. 

306-468 

Twice  do  X  Moon's  mass,   =     7-51744          a.  c. 
Lunar  precession,  0". 09424  18 

E  011  perigee,  0".01253  718 

=      d  D      (0-00000  02643  03) 
2fi  _i_  (1  _  C2)          =  1-00003  14038  5  = 


LCXAH     APS1DKS.  i> 

The,  value  of  «  is,  however,  a  direct  function  of  the  perturbation. 
We  obtain  it  as  follows: 

(1  —  <-2)'-r-  «  97-4479980098 

Syn.  .i.   sid.   period  of    D  0-03376  49993' 

=        *-  -4-    /2  97-48176  30091' 


Whence        e          =  0-0548997758  98.73957057 

«2        =  -0030139854  97-47914/11416 

e4        =  -0000090841  94-9582823 

gC        =  -0000000274  92-4374 

e8        =  -00000000008  89-92t 

p  0-9969860146  9-9986890662 

Then,  for  the  averages  on  radius  vector  we  have: 

(r  4-  »)»•        =  1-00904  53630  5'  0-00391  06910  0 

(r  4-  a)4         =  1-01508  69601  8  0-00650  44721  7 
Also  for  the  inclination  we  have: 
8  «a  _:_  3         = 


_:_  8     __     _|_         339627  /'    '      —    -   -  —  ^4-  0"804 

sin2    y'  =  0-00804069057  97-90529335 

sin4    y    ^    =  .  V6  46^27  \      9^105867 

sin8    y  42  91-6212 

When   n   is  %$n   even   power,  the    average  value    of    sin"    y    is 
\  (n  —  1  )  .  (n  —  2)  .  .  .'.  ($n  +  1) 
•.  2«(1    .    2    ;    3    .    4    .    .    .    !*  ._  Jw) 
Giving      ( 1  4-  2  )  for  sin2  ;  (3^-8)  for  sin4  ; 

(5  4-  16)  for  sin6;  (35  4-  128)  for  sin*;  etc. 

Cos2  ]>  's  latitude  =  1  —  sin2  y  sin2  longitude.  Hence  we  get 
the  following  values,  not  for  the  latitude  at  any  particular  point 
but  the  average  cos,  cos2,  etc.,  of  the  D 's  latitude: 

cos    lat.  =1  —  —  sin2  y  —  — -  sin4  y  — sin6  y  — ' —  sin  ®  y . 

4  64  256  16384 

cos2  lat.  =  1  —  J  sin2  y. 

395  105 

cos3  lat.  =  1  —  -  sin2  y  4-  —  sin4  y  4- sin6  y  4- sin8  y. 

4  64  256  16384 

cos4  lat.  =  1 —  sin2   y  -f-    f  sin4  y. 
These  relations  give  us: 

cos*   lat.          =         0-99397  85840' :  9-99737  70272 

cos4    lat.  =         0-9919835542:  9-9965044722 


0  MOTION    OF    THK 

And  these  multiplied  into  the  average  values  of  ra  and  r^  give 
the  average  third  and  fourth  powers  of  the  projection  of  r  on 
the  plane  of  the  ecliptic:  , 

It  is  important  to  note  that  the  sum  of  the  cube  cosines  for 
an  inclination  of  5°  8'  40".  61  9  is  equal  to  that  for  a  medial  value 
of  0".804  less;  so  that  our  computation  gives  us  5°  8'  39".  81  5  ±  5f. 
This  corresponds  precisely  to  the  HANSENIAN  value  of  5°8/39//.9G 
corrected  by  the  —  0".15  which  NEWCOMB  deduced  from  a  dis- 
cussion of  the  Greenwich  and  Washington  observations  from  18G2 
to  1874. 

If  £  be  the  )>  's  distance  divided  by  that  of  ©  ,  and  taking 
the  parallaxes  as  equal  to  3422".  75  and  8".  794,  we  have 
£2  _  0-000006601803;  and  the  value  of  3m  -f-  2  must  be  multi- 

9  15 

plied  into  (1  -j-  _  £2,  etc.)  and  (  1  +  _  £2,  etc.)  for  the  perturba- 

8  8 

tive  series  in  the  direction  of  r  and  perpendicular  thereto. 

For  the  effect  due  to  the  "variation,"  let  1  +  x  an(l  1  —  x 
represent  the  semi  -axes  of  the  ellipse,  the  longer  axis  being  in 
quadratures  and  the  other  in  the  syzigies.  Let  w  be  the  mean 
angular  distance  from  the  direction  of  the  minor  -axis  of  this 
ellipse.  Then  if  r0  denote  the  distance  from  the  centre  to  any 
point  in  the  circumference,  we  have,  by  comparison  of  the  ellipse 
with  its  circumscribing  circle: 
r2  —  sin2  (  w  +  dw  )  .  (1  +  x  )2  +  cos2  (  w  +  dw  )  .  (  1  —  x  )-; 

=     1  4~  a?2  —  2x  cos  2w  -f-  4Q  cc2  sin2  2w} 

if    qx     denote   the   maximum   perturbation    in    longitude    in    the 
average  orbit  —  that  which  gives  unequal  areas  in  equal  times. 

Now,  1  -jr  x2  =  a2,  if  a0  be  the  radius  of  the  circle  of  equal 
area  that  would  have  been  described  in  the  absence  of  compres- 
sion; because  (I  -\-  x)  .  (I  —  x)  =  1—  x2.  Hence 

•    ro  ~^~   ao      —      1  —  2cc  cos   2w  +  4Qx2  sin2  2w, 
if    there    were    110    change    of    area;     and    becomes 


+         ,      .   (1    _    2x  cos  2w  +   4Qic2  sin2  2  w  ) 


on    account    of   solar    perturbation    on   aQ.      'From  this  we  have: 
r>  =    (  1    +       )    -   (  1    +   |  *2  +   3  ^  +  |  Q^4) 
4       1   +   2x2+   4QX2+   6  Q2X4). 


LUNAR    APSIDES. 


The   numerical  values  are  as  follows: 


(  syn  4-   sid  )2  —  1         = 
=          4  x  -4-   (  1  +  *  )- 


0-1682344223 


1-7750601269' 
97-4791411416 

9-2259148612 
98-48011  6130 


which  is  the   square  of   the  average   eccentricity  in  the  hypotheti- 
cal orbit  described  by  the  moon   once  in  each  synodical  lunation. 


1  +  X 

I    -   X 

(1  4-  x)  4-  (1  - 
1  +  (*2  -5-  2) 


-0076681608 


•00005  88006  9 
=  tan  (45°+  1581".633)     = 


's  daily  motion   4- 
402".857 


average   cos 


add 


1581".633 


=   Variation;          = 


1984".490 
Syn  4-   sid 
2144".934; 
And   Q  = 

Taking  the  logarithms,  we  have: 

For  r3 

Function  of   e  0-00391  06910 

y  9-99737  70273 

x  0-00005 14323 

£  0-0000032255 


(log) 


(logs.) 
and  the 

numbei-s  are 


The  logarithm  of  the  sum 

(3  4-  2)  m 

Solar;     ( «v  4-  r,)3, 

Earth  perturbation, 

PI anetary  perturbation, 


0-0013423761 


97-88469  12105 
0-00331  75364 
9-9966569260 

95-7693824210 
0-0066606104 
0-00001  27682 

97-92511  926 
4-68003  181 
2-60515  107 

3-29764  88 
0-03376  50 
3-33141  39 
9-40603472' 

For  r4 

0-00650  32488 

9-99650  44722 

0-0000941089' 

53758 

0-0031072057' 


1-0030957170 
1-0071802610 
2-0102759780 


-01683252457 


0-3032556830' 

97-9226965831' 

0-0001837199 

0-00001  36383 

9-99999  96358 

98-2261492602 


THE     LUNAR    APSIDES. 

1  —  c  -00845198033  97-9:269584777 

D  4-67609  78999* 

(1  —  c)          =  400".918759264  2-6030563777, 

(The  planetary  perturbation  is  that  adopted  by  HILL  in  his 
tables  of  Venus.  It  is  what  LAPLACE  terms  the  "indirect"  per- 
turbation; being  that  clue  to  the  enlargement  of  the  earth's  radius 
vector  by  planetary  action,  which  lessens  the  solar  disturbing 
force.  The  direct  planetary  perturbation  is  neglected,  being 
infmitessimal  as  between  the  earth  and  moon.) 

This  result  is  identical  with  the  value  of  the  perigeal  motion 
which  NEWCOMB  has  obtained  from  a  discussion  of  the  eclipses 
of  2500  years  preceding  the  present  century.  The  difference 
between  the  two  is  less  than  one  part  in  100,000,000.  Hence 
the  problem  is  completely  solved. 

The  following  is  the  resulting  value  of  the  daily  motion  of  g, 
the  mean  anomaly:  47033".97147  44 

NEWCOMB,  47033".97147 

HASSEN;  (Tables  D  ),  47033".97227 

If  any  one  should  object  to  our  deduction  of  the  values  of  e 
and  y  from  that  of  the  quantity  sought  he  is  respectfully  referred 
to  the  top  of  page  174  of  Loomis'  Practical  Astronomy,  with  the 
fact  that  a  comparison  of  the  rates  of  change  in  the  values  of 
the  quantities  shows  this  to  be  a  parallel  case  with  that  given 
by  LOOMIS  on  page  173.  It  is  not  necessary  to  our  result  to 
carry  out  the  logarithm  of  e2  to  ten  places;  but  I  think  there 
needs  be  no  doubt  in  the  future  in  regard  to  the  precise  values 
of  e,  y,  or  x  in  the  lunar  orbit.  Of  course  the  numerical  values 
of  these  quantities  are  slightly  reduced  since  the  beginning  of  the 
century  by  the  decreasing  eccentricity  of  the  earth's  orbit.  There 
is  still  room  for  a  possible  very  small  correction  to  the  assumed 
values  of '  sidereal  motion  of  the  sun  and  moon. 


E.  COLBERT, 

Formerly  Superintendent  Dearborn  Observatory. 


CHICAGO,  December  12,   1886. 


FERGUS  PRINTING  COMPANY,  CHICAGO. 


Photomount 
Pamphlet 

Binder 
Gay  lord  Bros. 

Makers 
Stockton,  Calif. 

PAT.  JAN.  21.  1908 


HOME  USE 

CIRCULATION  DEPARTMENT 
MAIN  LIBRARY 

This  book  is  due  on  the  last  date  stamped  below. 
1 -month  loans  may  be  renewed  by  calling  642-3405. 
6-month  loans  may  be  recharged  by  bringing  books 

to  Circulation  Desk. 
Renewals  and  recharges  may  be  made  4  days  prior 

to  due  date. 

ALL  BOOKS  ARE  SUBJECT  TO  RECALL  7  DAYS 
AFTER  DATE  CHECKED  OUT. 


Z.W619    AON 


NQV 


OCT  1 


MAR  i  3  1982      It)     LIBRARY  USE 


0     MAR      3 


LD21 — A-40m-5,'74 
(R8191L) 


General  Library 

University  of  California 

Berkeley 


U.C.  BERKELEY  LIBRARIES 


